On Exponents of Thickness in Geometry Rigidity Inequality for Shells

We study exponents of thickness in Frieseck-James-M\"uller's inequalities for shells. We derive the following results: (a) the exponent of thickness $\mu(S)\leq15/8$ if the middle surface $S$ is parabolic; (b) the exponent of thickness $\mu(S)\leq11/6$ if the middle surface $S$ is a minimal surface with negative curvature; (c) the exponent of thickness $\mu(S)\leq11/6$ if the middle surface $S$ is a ruled surface with negative curvature. The exponents of thickness in Frieseck-James-M\"uller's inequalities for thin shells represent the relationship between rigidity and thickness $h$ of a shell when the large deformations take place, i. e., the rigidity of the shell related to the thickness $h$ is $$Ch^{\mu(S)}.$$ Thus the above results of $\mu(S)<2$ show that those shells are strictly more rigid than plates since $\mu(S)=2$ for plates. Moreover, we present another result which shows that when $\mu(S)<2,$ any $W^{2,2}$ isometry of the middle surface is rigid.